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Comparative analysis of critical regions: The renormalized quark-meson model under Polyakov loop, quark back-reaction, and vector interaction effects

T0 review · 1 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Consistently renormalized quark vacuum loops produce large, smooth, roughly symmetric critical regions around the quark-meson model's critical end point, unlike the pinched contours from curvature-mass parameter fixing.

desk verdict New susceptibility-contour maps for on-shell RQM/RPQM are worth a referee's time, but the central contrast with curvature-mass models rests on one renormalization-scheme choice that gets no sensitivity test. read the letter →

arxiv 2512.13398 v2 pith:6LC7IU4J submitted 2025-12-15 hep-ph

Akanksha Tripathi , Suraj Kumar Rai , Vivek Kumar Tiwari This is my paper · ORCID
classification hep-ph
keywords quark-mesonmodelPolyakovloopcriticalendpointtricriticalquarknumbersusceptibilityon-shellrenormalizationchiralphasetransitionbackreaction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the size and shape of the critical region around the QCD critical end point (CEP) depend heavily on how quark vacuum fluctuations are renormalized. Using an on-shell renormalization scheme, where meson masses are matched to their physical pole masses, the authors find that the CEP acquires a large, smooth, roughly symmetrical critical region—measured by contours of the normalized quark number susceptibility—instead of the narrow, pinched contours obtained when parameters are fixed by curvature masses. They also argue that at a scalar-meson mass of 500 MeV, the tricritical point lies inside the Rq=2 critical contour, so tricritical physics can influence the critical fluctuations seen around the CEP. The upshot: model predictions for where CEP signals should appear in heavy-ion experiments are sensitive to the vacuum-fluctuation treatment.

What carries the argument

The central object is the on-shell renormalized vacuum effective potential, built by matching MS-scheme counterterms to on-shell pole masses of π, K, η, η′, and σ, with the renormalization scale Λ0 fixed so the vacuum minimum does not shift from the unrenormalized model. The diagnostic is the ratio Rq = χq/χq^{free} of the quark number susceptibility to the free-quark-gas value, drawn as contours (Rq=2, 3, 5) in the μ-T plane. This ratio converts the CEP's divergent susceptibility into a finite-size, experimentally relevant 'critical region' whose shape and extent are compared across model settings.

What would settle it

Calculate the same susceptibility contours with a different, equally plausible renormalization condition (e.g., a different Λ0-fixing condition or matching at a different scale) and check whether the Rq=2 spreads change by more than ~10 MeV; if they do, the claimed 'large, smooth critical region' is an artifact of the scheme. Alternatively, a direct lattice QCD determination of the CEP's location and the phase-boundary curvature at finite density could falsify the RQM/RPQM predictions, which place the CEP near (265,39) MeV for mσ=500 or (243,37) MeV for mσ=400 in the physical-point parametriza

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Extended reading notes

Core claim

Consistent on-shell renormalization of the 2+1-flavor quark-meson model—counterterms matched to pole masses, scale Λ0 fixed by Eq. (13)—strengthens the 't Hooft coupling, weakens chiral symmetry breaking, and shifts the critical end point to larger μ, lower T than curvature-mass parametrizations. Contours of normalized quark number susceptibility Rq=χq/χq^{free} then envelop the CEP in large, smooth, roughly symmetric regions (Rq=2 spans 50.2 MeV in T, 58.3 MeV in μ for mσ=500 MeV), versus the narrow, pinched contours of QMVT/PQMVT models. For mσ=500 MeV the tricritical point sits inside the Rq=2 contour, so it can influence CEP fluctuations; for mσ=400 MeV it lies outside. Quark back-reacti

Load-bearing premise

The load-bearing premise is that the on-shell renormalization condition—fixing the scale Λ0 by Eq. (13) and matching counterterms to pole masses—is the correct and unique way to treat quark vacuum fluctuations; a different scheme would change the size and position of the predicted critical regions.

Editorial extensions

If this is right

  • If the on-shell treatment is correct, heavy-ion beam-energy-scan searches should expect a broad, rounded critical region extending tens of MeV in both temperature and chemical potential around the CEP, rather than a thin sliver.
  • For mσ=500 MeV, the tricritical point's location inside the Rq=2 contour means the CEP's critical fluctuations inherit tricritical influence, altering the expected scaling of higher-order cumulants.
  • The quark back-reaction in the PolyLog-glue Polyakov potential changes the shape of the critical region (rounder, broader in T), so modeling of the deconfinement transition matters for CEP phenomenology.
  • The RPQM model's CEP, placed at μ_B≈690–760 MeV and T≈71–95 MeV, lies closer to recent lattice and functional-renormalization-group estimates than the curvature-mass PQMVT model, suggesting the on-shell scheme is the more reliable effective-model benchmark.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial note: the abstract announces a vector-interaction analysis in which the CEP and first-order line survive up to g_ω=2.79, but the body text as provided contains no such section; that claim would need its own derivation before being used for compact-star phenomenology.
  • The paper itself cautions that two-decimal CEP coordinates reflect numerical binning, not physical precision—a useful reminder that the critical-region extent, not the nominal point, is the robust object.
  • A natural test: compute the same Rq contours in a functional renormalization group treatment of the same model to see whether the on-shell scheme's broad regions survive beyond mean-field.
  • If the TCP influence is real for mσ=500 MeV, the beam-energy-scan's net-proton cumulant ratios could show non-monotonicity from the combined O(4)-to-Z(2) crossover—an effect the paper maps but does not compute in detail.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper computes phase diagrams and quark-number-susceptibility contours (Rq = 2, 3, 5) around the critical end point in a 2+1 flavor quark-meson model with on-shell renormalized quark vacuum fluctuations (RQM), with and without Polyakov loop enhancement (RPQM), for mσ = 400 and 500 MeV. It reports that the consistently on-shell renormalized treatment yields larger, smoother, and more symmetric critical regions than the curvature-mass QMVT/PQMVT scheme of Ref. [118], and that for mσ = 500 MeV the tricritical point lies inside or on the Rq = 2 contour, implying TCP influence on critical fluctuations around the CEP. Light chiral limit phase diagrams are constructed using large-Nc ChPT inputs, and vector-interaction extensions are briefly discussed. The abstract explicitly cautions that the two-decimal CEP coordinates reflect numerical binning, not physical precision.

Significance. If the central claim holds, the paper would establish that the choice of meson-mass renormalization scheme is not a minor technical detail: it qualitatively changes the size and shape of the critical region around the CEP in a widely used effective model. The systematic comparison across QM/RQM/Log-RPQM/PolyLog-glue-RPQM and against Ref. [118] is useful, as is the light-chiral-limit TCP analysis. The paper is unusually candid in the abstract about binning artifacts in reported coordinates. However, no code or data archive is provided, and the central numerical results are imported from the authors' prior on-shell renormalization papers [141,142], so independent verification is currently not possible from the manuscript alone.

major comments (1)
  1. [Sec. II C and Table III] The light chiral limit results use ChPT inputs from Refs. [61–63] and the parameters are then used to locate the TCP. The paper reports that the TCP lies on/inside the Rq = 2 contour for mσ = 500 MeV but outside for mσ = 400 MeV. This proximity claim is potentially interesting, but it depends on both the on-shell scheme and the ChPT-determined fπ, fK, mη, mη′ in the chiral limit. Given the scheme sensitivity flagged above, the TCP-insertion conclusion should be revisited in the same sensitivity study.
minor comments (4)
  1. [Section IV] Typo: 'Rq = 2, 35, are drawn' should read 'Rq = 2, 3, 5'. Similar numerical typos appear in other places (e.g., 'mπ = 0, 35' and '(µCEP,T CEP)=(243.12.37.03)').
  2. [Fig. 4] The axis labels are labeled 'Tr (MeV)' and 'µr (MeV)', but Tr and µr are dimensionless ratios. Please correct the units.
  3. [Sec. III B] The term 'under no sea mean field approximation' should be 'standard mean field approximation (s-MFA)'. Also, the notation Rq is used for the ratio of susceptibilities, but the text sometimes writes 'Rq = 2, 3 5' with missing comma.
  4. [Eqs. (41)-(43)] The definition of χq_free as the massless free quark gas value is clear, but the contours of Rq = 2, 3, 5 are not derived from any universality or scaling argument. It would be helpful to state explicitly that these are heuristic measures of critical-region extent, not model-independent quantities.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: CEP/TCP and Rq contours are computed outputs, not fitted inputs; the on-shell scheme is a stated modeling assumption rather than a self-referential prediction.

full rationale

The paper's parameter fixing is standard and external: at the physical point, m_pi, m_K, M_eta, m_sigma, f_pi, f_K and the Yukawa coupling g determine the QM/RQM couplings (Table II), while the light-chiral-limit inputs follow from large-Nc ChPT with L8, v31 and v02 fixed to reproduce the physical M_eta [63]. The CEP, TCP and Rq=2/3/5 contours are then obtained by minimizing the grand potential and differentiating with respect to mu; none of these outputs is fed back into the parameter determination. Equation (13), which fixes Lambda_0 by requiring that the RQM vacuum minimum does not shift from the QM minimum, is an explicit renormalization-scheme condition imported from the authors' prior papers [141,142]; this is a model assumption whose sensitivity is not tested, but it is not a hidden fit and does not make the critical-region claim equivalent to its inputs. The statement that on-shell pole masses m_eta and m_eta' are reproduced after self-energy corrections is a self-consistency check of the chosen OS scheme, not a prediction derived from the scheme. The heavy reliance on Refs. [63,141,142] is cumulative use of prior peer-reviewed derivations; no uniqueness theorem is invoked to forbid alternatives, and no ansatz is passed off as externally established. Hence no step in the derivation chain reduces by construction to its own input; the scheme-dependence concern belongs to robustness/sensitivity rather than circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on the on-shell renormalized effective potential from the authors' previous works, on the Polyakov-loop potential parametrizations from the literature, and on the chosen sigma mass and T0 values. No new particle, force, or conserved quantity is introduced. The mean-field and ChPT extrapolation assumptions are standard for this class of model, but the on-shell scheme is specific to this research line.

free parameters (4)
  • sigma meson mass m_sigma = 400 and 500 MeV
    Chosen by hand as model inputs; controls the curvature of the effective potential and hence the CEP/TCP positions and contour shapes. The paper does not fit these values.
  • Yukawa coupling g = not stated in visible text
    The quark-meson coupling g enters all quark masses (m_u = g x/2, m_s = g y/sqrt(2)) and is required to fix the constituent quark mass scale. Its value is not listed in Table II, so the calculation depends on an implicit parameter from earlier works.
  • Polyakov scale T0 (and T_glue^c) = 187 MeV for 2+1 flavors; 270 MeV for pure gauge
    Input from lattice/FRG analyses (Refs. [41,46]); the paper sets T0 = T_glue^c = 187 MeV for the 2+1-flavor Polyakov-loop potentials. The comparison with Ref. [118] uses T0=270 MeV. The main conclusions depend on these choices.
  • PolyLog-glue temperature mapping coefficient = 0.57
    Taken from Ref. [46] to map the pure-gauge temperature to the unquenched glue temperature (Eq. 31). This coefficient controls how strongly quark back-reaction modifies the Polyakov-loop potential.
assumptions (5)
  • domain assumption Mean-field treatment of meson fields: only quark/antiquark thermal and vacuum fluctuations are included; meson fields are treated classically.
    Invoked throughout Section II; the grand potential in Eqs. (6), (15), (35) contains no meson fluctuation contribution beyond the tree-level potential.
  • ad hoc to paper On-shell renormalization scheme: pole masses of mesons are the physical masses, and counterterms are matched between MS and on-shell schemes as developed in Refs. [141,142].
    The renormalized effective potential Eq. (14) and the finite corrections m2_FIN, hxFIN, etc. are taken as given from the authors' previous works. The entire comparison between RQM and older QMVT results rests on this scheme.
  • domain assumption Large-N_c U(3) chiral perturbation theory inputs are used to extrapolate f_pi, f_K, m_eta, m_eta' to the light chiral limit (m_pi=0, m_K=496 MeV), with L8, v31, v02 fixed to reproduce M_eta at the physical point.
    Section II C, Eqs. (38)-(40). The TCP locations and the m_pi=0 phase diagrams depend on this ChPT extrapolation.
  • domain assumption The logarithmic and PolyLog-glue forms of the Polyakov-loop potential, with the T0(T_glue^c)=187 MeV mapping, faithfully capture confinement-deconfinement and quark back-reaction effects.
    Section II A: the Log potential (22) and PolyLog-glue potential (25)-(31) are imported from Refs. [40,46,47]. The paper's central back-reaction comparison assumes this mapping is valid.
  • standard math The quark number susceptibility ratio R_q = chi_q / chi_q^free identifies the critical region, with R_q=2,3,5 contours representing enhanced critical fluctuations.
    Section III, Eqs. (41)-(43). This is a standard diagnostic in the QCD model literature and is not an invented quantity.

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Pith. "Pith review of Comparative analysis of critical regions: The renormalized quark-meson model under Polyakov loop, quark back-reaction, and vector interaction effects." pith.science (2026). https://pith.science/paper/6LC7IU4J

@misc{pith2026251213398,
  author       = {Pith},
  title        = {Pith review of: Comparative analysis of critical regions: The renormalized quark-meson model under Polyakov loop, quark back-reaction, and vector interaction effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6LC7IU4J}},
  note         = {Machine review of arXiv:2512.13398}
}
abstract

The critical regions enveloping the critical end point (CEP) in the $\mu$-$T$ plane are mapped by computing the contours of normalized quark number susceptibility within the on shell renormalized 2+1 flavor quark-meson (RQM) and Polyakov loop enhanced renormalized Polyakov quark meson (RPQM) models for $m_\sigma = 400$ and 500 MeV.The apparent precision for the results of CEP coordinates merely reflects numerical binning of two decimal places rather than the effect of including full thermal and vacuum quantum fluctuations. The renormalized 't Hooft coupling c becomes substantially stronger in the RQM model when the meson self energies due to quark loops are computed using the pole masses of mesons and parameters are fixed on shell in Ref [143] after a consistent treatment of quark one loop vacuum fluctuations while the light and strange chiral symmetry breaking strengths also become weaker. We evaluate the impact of these novel features on critical fluctuations. Furthermore, the improved PolyLog glue form of the Polyakov loop potential from Ref [46] is employed to isolate the effects of the quark back reaction on critical fluctuations, and the results are contrasted against back reaction free outcomes obtained using the logarithmic potential. Utilizing inputs from large $N_c$ standard chiral perturbation theory, phase diagrams are also computed in the light chiral limit ($m_\pi = 0$), quantifying the proximity of the tricritical point (TCP) to the CEP. The critical regions from the RQM/RPQM models are compared with those reported in Ref [120], where curvature masses are used for parameter fixing. Phase diagrams incorporating vector interactions in the RQM/RPQM model reveal that the CEP and first-order transition survive up to a robust coupling of $g_\omega = 2.79$, rendering them highly relevant for compact star equations of state and astrophysical phenomenology.

Figures

Figures reproduced from arXiv: 2512.13398 by the authors.

Figure 1
Figure 1. FIG. 1. Temperature variations of the non-strange condensa [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The RQM model phase diagrams for the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. (b). The Polyakov loop potential with no quark back reaction in the Log RPQM model, generates compression in the temperature direction of the critical region which gets large elongation in the chemical poten￾tial direction, therefore the Rq = 2 susceptibility contour looks pinched near the CEP in the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The phase diagrams for the physical point and light ch [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: (a) is 19.4 MeV in the temperature direction (12.9 MeV above and 6.5 MeV below the TCEP on the T axis) and 51.3 MeV in the chemical potential direction (35.9 MeV lower and 15.4 MeV higher than the µCEP on the µ axis). For the Rq = 3 contour of the QM model, the size is…
Figure 6
Figure 6. Figure 6: (a) for the mσ = 400 MeV, are smaller than the corresponding contour sizes for the mσ = 500 MeV in the sequential order of (10.1,9.1 and 6.3) MeV on the T axis and (18.2,16.2 and 10.6) MeV on the µ axis. The chemical potential direction stretching of the Log RPQM model…
Figure 7
Figure 7. Figure 7: FIG. 7. The RPQM-I and RPQM-II model phase diagrams for the li [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. With the [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

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Works this paper leans on

157 extracted references · 5 linked inside Pith

  1. [118]

    E. S. Fraga, L. F. Palhares, and M. B. Pinto, Phys. Rev. D 79, 065026 (2009)

    2009

  2. [1]

    Cabibbo and G

    N. Cabibbo and G. Parisi, Phys.Lett.B 59 ,67-69(1975)

    1975

  3. [2]

    PQMVT model CEP is placed at significantly higher TCEP = 90 MeV whereas the CEP in both the QMVT and PQMVT models gets located at larger chemical po- tentials around 280–285 MeV [ 118]. The PQMVT model contour of a successively decreasing width below its TCEP, has a larger neck like narrow structure because the curvature of the first order transition line b...

  4. [3]

    Muller, Rep

    B. Muller, Rep. Prog. Phys. 58, 611 (1995)

    1995

  5. [4]

    The curvatures of the crossover and first order transition lines in the Fig

    Divid- ing the temperature T by the highest temperature T χ c of the chiral crossover transition on the temperature axis at µ = 0, one gets the Tr = T T χ c and when the chemical po- tentialµ is divided by the highest chemical potential µ0 at which first order phase transition on the µ axis occurs at T = 0, one gets the µr = µ µ0 . The curvatures of the cr...

  6. [5]

    D. H. Rischke, Prog. Part. Nucl. Phys. 52, 197 (2004)

    2004

  7. [6]

    A.Ali Khan et al. Phys. Rev. D 64 , 074510 (2001)

    2001

  8. [7]

    Digal, E

    S. Digal, E. Laermann and H. Satz, Eur. Phys. J. C 18 , 583 (2001) . [8]

    2001

Show all 157 references
  1. [8]

    to the thermal contributions of quarks-antiquarks in the presence of the Polyakov loop potential as the fol- lowing. ΩP QM(T,µ ;x,y, Φ, ¯Φ) = U (x,y ) + Ωq ¯q(T,µ ;x,y, Φ, ¯Φ) (36) The temperature T and chemical potential µ depen- dence of the non-strange condensate x, strange...

  2. [9]

    C. R. Allton, M. Doring, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E Laermann and K. Redlich, Phys. Rev. D 71 , 054508 (2005)

    2005

  3. [10]

    in the quark-antiquark contribution to the total effective potential, gets modified by the Polyakov loop potential. ΩT,µ q ¯q (x,y, Φ, ¯Φ) = −2Nc ∑ q ∫ d3p (2π)3T [ lng+ f + lng− f ] .(32) The Ω T,µ q ¯q (x,y, Φ, ¯Φ) = Ω q ¯q(T,µ ;x,y, Φ, ¯Φ) and new g± f being different from the...

  4. [11]

    L. D. McLerran and B. Svetitsky, Phys. Rev. D 24, 450 (1981) ; B. Svetitsky, Phys. Rep. 132, 1 (1986)

    1981

  5. [12]

    Meyer-Ortmanns, Rev

    H. Meyer-Ortmanns, Rev. Mod. Phys. 68, 473 (1996)

    1996

  6. [13]

    Cheng et al., Phys

    M. Cheng et al., Phys. Rev. D 77 , 014511 (2008)

    2008

  7. [14]

    is added to the thermal contributions of quarks-antiquarks in the presence of the Polyakov loop potential. ΩRP QM (T,µ ;x,y, Φ, ¯Φ) = Ω RQM vac (x,y ) + Ωq ¯q(T,µ ;x,y, Φ, ¯Φ) (35) When quark one-loop vacuum contribution is dropped in the s-MF A,one gets the PQM model grand th...

  8. [15]

    Bazavov et al., Phys

    A. Bazavov et al., Phys. Rev. D 80, 014504 (2009)

    2009

  9. [16]

    Fodor, S

    Z. Fodor, S. D. Katz, and K. K. Szabo, Phys. Lett. B 568 , 73 (2003)

    2003

  10. [17]

    Karsch, Lect

    F. Karsch, Lect. Notes Phys. 583, 209 (2002)

    2002

  11. [18]

    Karsch, J

    F. Karsch, J. Phys. G 31 , S633 (2005)

    2005

  12. [19]

    Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, Phys. Lett. B 643 , 46 (2006)

    2006

  13. [20]

    Cheng et al., Phys

    M. Cheng et al., Phys. Rev. D 74 , 054507 (2006)

    2006

  14. [21]

    Fejos, A

    G. Fejos, A. Hosaka, Phys. Rev. D 94, 036005 (2016)

    2016

  15. [22]

    Langelage, S

    J. Langelage, S. Lottini, and O. Philipsen, J. High Energy Phys. 02 (2011) 057

    2011

  16. [23]

    Fejos, A

    G. Fejos, A. Hosaka, Phys. Rev. D 98, 036009 (2018)

    2018

  17. [24]

    Ejiri et al., Phys

    S. Ejiri et al., Phys. Rev. D 80, 094505 (2009)

    2009

  18. [25]

    (27) The summary of the parameters is given in the Table I

    are defined as : ai(T ) = a(i) 0 +a(i) 1 ( T0 T ) +a(i) 2 ( T0 T ) 2 1 +a(i) 3 ( T0 T ) +a(i) 4 ( T0 T ) 2 (26) b(T ) = b0 ( T0 T ) b1[ 1 −eb2( T0 T ) b3 ] . (27) The summary of the parameters is given in the Table I. TABLE I. Parameters of the PolyLog Polyakov-loop potential h...

  19. [26]

    M. G. Alford, A. Schmitt and K .Rajagopal, Rev. Mod. Phys. 80, 1455 (2008)

    2008

  20. [27]

    Fukushima, and T

    K. Fukushima, and T. Hatsuda, Rep. Prog. Phys. 74, 014001 (2011)

    2011

  21. [28]

    Fejos, Phys

    G. Fejos, Phys. Rev. D 92, 036011 (2015)

    2015

  22. [29]

    J. T. Lenaghan, Phys. Rev. D 63 , 037901 (2001)

    2001

  23. [30]

    D 96, 016009 (2017)

    Fabian Rennecke, Bernd-Jochen Schaefer, Phys, Rev. D 96, 016009 (2017)

    2017

  24. [31]

    Pedro Costa, M. C. Ruivo, C. A. de Sousa and Yu. L. Kalinovsky, Phys. Rev. D 71 , 116002 (2005)

    2005

  25. [32]

    Fejos, A

    G. Fejos, A. Patkos, Phys. Rev. D 105, 096007 (2022)

    2022

  26. [33]

    ’t Hooft, Phys

    G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976)

    1976

  27. [34]

    B. J. Schaefer and M. Wagner, Phys. Rev. D 79 , 014018 (2009)

    2009

  28. [35]

    and Eq. ( 36) . ∂ΩRP QM/P QM ∂x = ∂ΩRP QM/P QM ∂y = ∂ΩRP QM/P QM ∂Φ = ∂ΩRP QM/P QM ∂ ¯Φ = 0 (37) The 2+1 flavor RPQM and PQM model phase diagram with Log form and PolyLog-glue form of the Polyakov loop potential, different thermodynamic quantities and critical end point have bee...

  29. [36]

    Meyer-Ortmanns and B

    H. Meyer-Ortmanns and B. J. Schaefer, Phys. Rev. D 53 , 6586 (1996)

    1996

  30. [37]

    J. T. Lenaghan, D. H. Rischke and J. Schaffner-Bielich, Phys. Rev D 62 , 085008 (2000) . J. T. Lenaghan, D. H. Rischke, J. Phys. G 26 , 431 (2000)

    2000

  31. [38]

    Pedro Costa, M. C. Ruivo, C. A. de Sousa and Yu. L. Kalinovsky, AIP Conf. Proc. 775, 173–181 (2005) . 21

    2005

  32. [39]

    Ratti, M

    C. Ratti, M. A. Thaler, and W. Weise, Phys. Rev. D 73, 014019 (2006)

    2006

  33. [40]

    Fukushima, Phys

    K. Fukushima, Phys. Rev. D 77 , 114028 (2008); 78, 039902(E) (2008)

    2008

  34. [41]

    A. M. Polyakov, Phys. Lett. B 72 , 477 (1978)

    1978

  35. [42]

    Svetitsky and L

    B. Svetitsky and L. G. Yaffe, Nucl. Phys. B 210, 423 (1982)

    1982

  36. [43]

    Banks and A

    T. Banks and A. Ukawa, Nucl. Phys. B 225, 145 (1983)

    1983

  37. [44]

    R. D. Pisarski, Phys. Rev. D 62 , 111501(R) (2000)

    2000

  38. [45]

    Fukushima, Phys

    K. Fukushima, Phys. Lett. B 591, 277 (2004)

    2004

  39. [46]

    Layek, A

    B. Layek, A. P. Mishra, A. M. Srivastava and V. K. Tiwari, Phys. Rev. D 73 , 103514 (2006)

    2006

  40. [47]

    P. M. Lo, B. Friman, O. Kaczmarek, K. Redlich, and C. Sasaki, Phys. Rev. D 88 , 074502 (2013)

    2013

  41. [48]

    Roessner, C

    S. Roessner, C. Ratti and W. Weise, Phys. Rev. D 75, 034007 (2007)

    2007

  42. [49]

    B. J. Schaefer, J. M. Pawlowski, and J. Wambach, Phys. Rev. D 76 , 074023 (2007)

    2007

  43. [50]

    B. J. Schaefer, M. Wagner, and J. Wambach, Phys. Rev. D 81 , 074013 (2010)

    2010

  44. [51]

    B. J. Schaefer, M. Wagner and J. Wambach, CPOD(2009)017, arXiv:0909.0289

    2009 arXiv

  45. [52]

    H. Mao, J. Jin, and M. Huang, J. Phys. G 37 , 035001

  46. [53]

    U. S. Gupta and V. K. Tiwari, Phys. Rev. D 81 , 054019 (2010)

    2010

  47. [54]

    L. M. Haas, R. Stiele, J. Braun, J. M. Pawlowski and J. Schaffner-Bielich, Phys. Rev. D 87 , 076004 (2013)

    2013

  48. [55]

    Asakawa and K

    M. Asakawa and K. Yazaki, Nucl. Phys. A 504 , 668, (1989)

    1989

  49. [56]

    Stiele and J

    R. Stiele and J. Schaffner-Bielich, Phys. Rev. D 93 , 094014 (2016)

    2016

  50. [57]

    T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer, Phys. Lett. B 696, 58 (2011)

    2011

  51. [58]

    T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer, Phys. Rev. D 88, 014007 (2013)

    2013

  52. [59]

    Bors´ anyi, Z

    S. Bors´ anyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szab´ o, J. High Energy Phys. 09 (2010) 73

    2010

  53. [60]

    Borsanyi, G

    S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg, C. Ratti and K. K. Szabo, JHEP 11, (2010) 077 . ; S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg and K. K. Sz- abo, Phys. Lett. B 730 (2014) 99-104

    2010

  54. [61]

    Bazavov et al., Phys

    A. Bazavov et al., Phys. Rev. D 85 , 054503 (2012) ; Phys. Rev. D 90, 094503 (2014)

    2012

  55. [62]

    R. D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984)

    1984

  56. [63]

    V. K. Tiwari, Phys. Rev. D 111, 114014 (2025)

    2025

  57. [64]

    Barducci, R

    A. Barducci, R. Casalbuoni, S. De Curtis, R. Gatto, and G. Pettini, Phys. Lett. B 231 , 463 (1989) ; Phys. Rev. D 41 , 1610 (1990)

    1989

  58. [65]

    Berges and K

    J. Berges and K. Rajagopal, Nucl. Phys. B 538 , 215 (1999)

    1999

  59. [66]

    Hatta and T

    Y. Hatta and T. Ikeda, Phys. Rev. D 67 , 014028 (2003)

    2003

  60. [67]

    Fujii, Phys

    H. Fujii, Phys. Rev. D 67 , 094018 (2003)

    2003

  61. [68]

    8 838-844, (2011)

    Fukushima, Kenji Physics of Particles and Nuclei Letters. 8 838-844, (2011)

    2011

  62. [69]

    Herrera-Sikl´ ody, J

    P. Herrera-Sikl´ ody, J. I. Latorre, P. pascual and J. Taron, Phys. Lett. B 419, 326 (1998)

    1998

  63. [70]

    Escribano, F

    R. Escribano, F. S. Ling, M. H. G. Tytgat, Phys. Rev. D 62, 056004 (2000)

    2000

  64. [71]

    G.Odyniec; RHIC Beam Energy Scan Program: Phase I and II, PoS CPOD 2013, 043(2013) and PoS CORFU2018 151 (2019)

    2013

  65. [72]

    V. K. Tiwari, Phys. Rev. D 112, 054046 (2025)

    2025

  66. [73]

    D. T. Son and M. A. Stephanov, Phys. Rev. D 70 , 056001 (2004) ; Y. Aoki, G. Androdi, Z. Fodor, S. D. Katz, and K. K. Szabo, Nature (London) 443, 675 (2006) ; S. Gupta,X. Luo, B. Mohanty, H. G. Ritter, and N. Xu, Science 332,1525 (2011)

    2004

  67. [74]

    M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Phys.Rev. D 60, 114028 (1999) ; Phys. Rev. Lett. 81, 4816 (1998)

    1999

  68. [75]

    Berdnikov and K

    B. Berdnikov and K. Rajagopal, Phys. Rev. D 61 , 105017 (2000)

    2000

  69. [76]

    Jeon and V

    S. Jeon and V. Koch, Phys. Rev. Lett. 85, 2076 (2000)

  70. [77]

    Ejiri, F

    S. Ejiri, F. Karsch, and K. Redlich, Phys. Lett. B 633 , 275 (2006)

    2006

  71. [78]

    Adams et al

    J. Adams et al. (STAR Collaboration), Nucl. Phys. A 757, 102 (2005) ; M. Ag- garwal et al. (STAR Collaboration), Phys. Rev. Lett. 105, 022302 (2010)

    2005

  72. [79]

    Mohanty, Nucl

    B. Mohanty, Nucl. Phys. A 830, 899c (2009) .; X. Luo, B. Mohanty, H. G. Ritter, and N. Xu, Phys. At. Nucl. 75, 676 (2012)

    2009

  73. [80]

    Liu et al

    C. Liu et al. 13th Int. Particle Acc. Conf. WEPOPT032

  74. [81]

    296, 01016, (2024) ; A

    Pandav, Ashish; EPJ Web Conf. 296, 01016, (2024) ; A. Pandav (STAR collaboration), plenary talk at CPOD 2024,

    2024

  75. [82]

    Stephanov, EPJ Web of Conferences 314, 00042 (2024)

    M. Stephanov, EPJ Web of Conferences 314, 00042 (2024)

    2024

  76. [83]

    Bzdak, S

    A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov, N. Xu, Phys. Rept. 853, 1 (2020)

    2020

  77. [84]

    L. Du, A. Sorensen, M. Stephanov Int. J. Mod. Phys. E 33 (2024) 07, 2430008

    2024

  78. [85]

    Stephanov, Phys

    M.A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009)

    2009

  79. [86]

    Athanasiou, K

    C. Athanasiou, K. Rajagopal, and M. Stephanov, Phys. Rev. D 82 , 074008 (2010)

    2010

  80. [87]

    Karthein, M.S

    J.M. Karthein, M.S. Pradeep, K. Rajagopal, M. Stephanov, Y. Yin, Equilibrium expectations for non- Gaussian fluctuations near a QCD critical point, in 21st International Conference on Strangeness in Quark Mat- ter 2024 arXiv:2409.16249 (nucl-th)

    2024 arXiv

  81. [88]

    M. A. Stephanov, Phys. Rev. Lett. 107, 052301 (2011)

    2011

  82. [89]

    Dunne et

    Xin An, Marcus Bluhm, Lipei Du and Gerald V. Dunne et. All Nuclear Physics A,1017,(2022)

    2022

  83. [90]

    Akamatsu Yukinao and Teaney Derek and Yan Fanglida and Yin Yi Phys. Rev. C 100, 044901 (2019)

    2019

  84. [91]

    Vovchenko, V

    V. Vovchenko, V. Koch and C. Shen, Phys. Rev. C 105, 014904 (2022)

    2022

  85. [92]

    X. An, G. Basar, M. Stephanov, H.U. Yee, Phys. Rev. Lett. 127, 072301 (2021) . Phys. Rev. C 108 , 034910 (2023)

    2021

  86. [93]

    Pradeep, K

    M. Pradeep, K. Rajagopal, M. Stephanov, Y. Yin, Freezing out fluctuations in Hy- dro+ near the QCD critical point, Phys. Rev. D 106 , 036017 (2022) ; M.S. Pradeep, M. Stephanov, Phys. Rev. Lett. 130, 162301 (2023)

    2022

  87. [94]

    Parotto, M

    P. Parotto, M. Bluhm, D. Mroczek, M. Nahrgang, J. Noronha-Hostler, K. Rajagopal, C. Ratti, T. Sch¨ afer, M. Stephanov, Phys. Rev. C 101 , 034901 (2020)

    2020

  88. [95]

    Kovacs and Zs

    P. Kovacs and Zs. Szep, Phys. Rev. D 75 , 025015 (2007)

    2007

  89. [96]

    D. Roder,J. Ruppert and D. H. Rischke, Phys. Rev. D 68 , 016003 (2003)

    2003

  90. [97]

    Fukushima,K

    K. Fukushima,K. Kamikado and B. Klein, Phys. Rev. D 83 , 116005 (2011) . 22

    2011

  91. [98]

    Grahl and D

    M. Grahl and D. H. Rischke, Phys. Rev. D 88 , 056014 (2013)

    2013

  92. [99]

    Jakovac, A

    A. Jakovac, A. Patkos, Z. Szep, and P. Szepfalusy, Phys. Lett. B 582 , 179 (2004)

    2004

  93. [100]

    Herpay, A

    T. Herpay, A. Patk´ os, Zs. Sz´ ep and P. Sz´ epfalusy, Phys. Rev. D 71 , 125017 (2005)

    2005

  94. [101]

    Herpay and Zs

    T. Herpay and Zs. Sz´ ep, Phys. Rev. D 74 , 025008 (2006)

    2006

  95. [102]

    Kov´ acs and Zs

    P. Kov´ acs and Zs. Sz´ ep, Phys. Rev. D 75 , 025015 (2007)

    2007

  96. [103]

    Scavenuius, A

    O. Scavenuius, A. Mocsy, I. N. Mishustin, and D. H. Rischke, Phys. Rev. C 64, 045202 (2001)

    2001

  97. [104]

    Kahara and K

    T. Kahara and K. Tuominen, Phys. Rev. D 78 , 034015 (2008) ; 80, 114022 (2009) ; 82, 114026 (2010)

    2008

  98. [105]

    E. S. Bowman and J. I. Kapusta, Phys. Rev. C 79 , 015202 (2009) ; J. I. Kapusta, and E. S. Bowman, Nucl. Phys. A 830 , 721C (2009)

    2009

  99. [106]

    Fejos, A

    G. Fejos, A. Patkos, Phys. Rev. D 82 , 045011 (2010)

    2010

  100. [107]

    Jakovac and Zs

    A. Jakovac and Zs. Szep, Phys. Rev. D 82 , 125038, (2010)

    2010

  101. [108]

    Ferroni, V

    L. Ferroni, V. Koch, and M. B. Pinto, Phys. Rev. C 82 , 055205 (2010)

    2010

  102. [109]

    Marko and Zs

    G. Marko and Zs. Szep, Phys. Rev. D 82 , 065021 (2010)

    2010

  103. [110]

    Therefore the mapping of the critical regions around the CEP of the RQM and RPQM model becomes important

    MeV and µB ∼ (420 − 650) ⇔µCEP ∼ (140 − 217) MeV given by the most recent results of different theoret- ical techniques [ 74]. Therefore the mapping of the critical regions around the CEP of the RQM and RPQM model becomes important. The temperature variations of the order param...

  104. [111]

    Gergely Fej˝ os and Andr´ as Patk´ os, Phys. Rev. D 112 ; 7, 076013 (2025)

    2025

  105. [112]

    Mocsy, I

    A. Mocsy, I. N. Mishustin, and P. J. Ellis, Phys. Rev. C 70 , 015204 (2004)

    2004

  106. [113]

    Schaefer and J

    B.-J. Schaefer and J. Wambach, Nucl. Phys. A 757 , 479 (2005)

    2005

  107. [114]

    Schaefer and J

    B.-J. Schaefer and J. Wambach, Phys. Rev. D 75 , 085015 (2007)

    2007

  108. [115]

    Halasz, A

    A. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanov, and J. J. M. Verbaarschot, Phys. Rev. D 58, 096007 (1998)

    1998

  109. [116]

    Skokov, B

    V. Skokov, B. Friman, E. Nakano, K. Redlich, and B.-J. Schaefer, Phys. Rev. D 82, 034029 (2010)

    2010

  110. [117]

    L. F. Palhares and E. S. Fraga, Phys. Rev. D 78, 025013 (2008)

    2008

  111. [119]

    L. F. Palhares and E. S. Fraga, Phys. Rev. D 82, 125018 (2010)

    2010

  112. [120]

    Gatto, M

    R. Gatto, M. Ruggieri, Phys. Rev. D 82, 054027 (2010)

    2010

  113. [121]

    A. J. Mizher, M. N. Chernodub, and E. S. Fraga, Phys. Rev. D 82, 105016 (2010)

    2010

  114. [122]

    Khan and L

    R. Khan and L. T. Kyllingstad, AIP Conf. Proc. 1343, 504 (2011)

    2011

  115. [123]

    U. S. Gupta, V. K. Tiwari, Phys. Rev. D 85, 014010 (2012)

    2012

  116. [124]

    Chatterjee and K

    S. Chatterjee and K. A. Mohan, Phys. Rev. D 85, 074018 (2012)

    2012

  117. [125]

    V. K. Tiwari, Phys. Rev. D 86, 094032 (2012)

    2012

  118. [126]

    Schaefer and M

    B.-J. Schaefer and M. Wagner, Phys. Rev. D 85, 034027 (2012)

    2012

  119. [127]

    J. O. Andersen and A. Tranberg, J. High Energy Phys. 08 (2012) 002

    2012

  120. [128]

    Kharzeev, K

    D. Kharzeev, K. Landsteiner, A. Schmitt, H.-U. Yee (Eds.),Strongly Interacting Matter in Magnetic Fields, Lecture Notes in Phys. 871,Spriner (2013)

    2013

  121. [129]

    V. K. Tiwari, Phys. Rev. D 88, 074017 (2013)

    2013

  122. [130]

    Weyrich, N

    J. Weyrich, N. Strodthoff, and L. von Smekal, Phys. Rev. C 92, 015214 (2015)

    2015

  123. [131]

    M. Drew, W. Weise, Phys. Rev. C 91, 035802 (2015)

    2015

  124. [132]

    Kov´ acs, Zs Sz´ ep, Gy Wolf, Phys

    P. Kov´ acs, Zs Sz´ ep, Gy Wolf, Phys. Rev. D 93, 114014 (2016)

    2016

  125. [133]

    Andreas Zacchi and J¨ urgen Schaffner-Bielich, Phys. Rev. D 97, 074011 (2018)

    2018

  126. [134]

    Andreas Zacchi and J¨ urgen Schaffner-Bielich, Phys. Rev. D 100, 0123024 (2019)

    2019

  127. [135]

    S. K. Rai and V. K. Tiwari, Eur. Phys. J. Plus 135:844, (2020)

    2020

  128. [136]

    Brandes, N

    L. Brandes, N. Kaiser, W. Weise, Eur. Phys. J. A 57:243, (2021)

    2021

  129. [137]

    Kobes, G

    R. Kobes, G. Kunstatter, and A. Reb- han, Phys. Rev. Lett. 64, 2992 (1990); Nucl. Phys. B 355, 1 (1991)

    1990

  130. [138]

    A. K. Rebhan, Phys. Rev. D 48, R3967 (1993)

    1993

  131. [139]

    Kajantie, M

    K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov Nucl. Phys. B 458, 90 (1996)

    1996

  132. [140]

    Carignano, M

    S. Carignano, M. Buballa and B-J Schaefer Phys. Rev. D 90, 014033 (2014)

    2014

  133. [141]

    Carignano, M

    S. Carignano, M. Buballa, and W. Elkamhawy, Phys. Rev. D 94, 034023 (2016)

    2016

  134. [142]

    J. O. Andersen, W. R. Naylor, and A. Tranberg, Rev. Mod. Phys. 88, 025001 (2016)

    2016

  135. [143]

    Adhikari, J

    P. Adhikari, J. O. Andersen and P. Kneschke, Phys. Rev. D 95, 036017 (2017)

    2017

  136. [144]

    Adhikari, J

    P. Adhikari, J. O. Andersen and P. Kneschke, Phys.Rev.D 96, 016013 (2017)

    2017

  137. [145]

    Adhikari, J

    P. Adhikari, J. O. Andersen and P. Kneschke, Phys.Rev.D 98, 074016 (2018)

    2018

  138. [146]

    A.Folkestad, J. O. Andersen, Phys.Rev.D 99, 054006 (2019)

    2019

  139. [147]

    S. K. Rai and V. K. Tiwari, Phys.Rev.D 105, 094010 (2022)

    2022

  140. [148]

    S. K. Rai and V. K. Tiwari, Phys.Rev.D 108, 074014 (2023)

    2023

  141. [149]

    V. K. Tiwari, Phys. Rev. D 108, 074002 (2023)

    2023

  142. [150]

    S. K. Rai and V. K. Tiwari, Phys. Rev. D 109, 034025 (2024)

    2024

  143. [151]

    Andersen and Mathias P

    Jens O. Andersen and Mathias P. Nødtvedt, arXiv:2506.02941

    arXiv

  144. [152]

    Hosein Gholami, Kurth Lennart, Ugo Mire, Michael Buballa, and Bernd-Jochen Schaefer, arXiv:2505.22542

    arXiv

  145. [153]

    D. A. Clarke, P. Dimopoulos, F. Di Renzo, J. Goswami, C. Schmidt, S. Singh, K. Zambello arXiv:2405.10196 [hep-lat]

    arXiv

  146. [154]

    Basar, Phys Rev C 110 , 015203, (2024)

    G. Basar, Phys Rev C 110 , 015203, (2024)

    2024

  147. [155]

    Hippert, J

    M. Hippert, J. Grefa, T. A. Manning, J. Noronha, J. Noronha-Hostler, I. P. Vazquez, C. Ratti, R. Rouge- mont, M. Trujillo. Phys. Rev. D 110 , 094006 (2024)

    2024

  148. [156]

    Y. Lu, F. Gao, Y.X. Liu, J.M. Pawlowski. Phys. Rev. D 110, 014036 (2024)

    2024

  149. [157]

    Juan Xiong, Meng Jin, and Jiarong Li Phys. Rev. C 83, 025204 (2011)

    2011

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